Continuous Waves and Stochastic Backgrounds

Long-lived gravitational waves occupy two complementary inference regimes. A continuous-wave search follows a deterministic phase trajectory so weak amplitude accumulates coherently. A stochastic search gives up the phase of each constituent source and instead measures correlations predicted for a random field. Both regimes demand long-baseline control of clocks, calibration, stationarity, spectral lines, trials, and computational convergence.

The observational examples in this chapter are pinned to 2026-08-18. The cited releases report upper limits rather than detections for the targeted, all-sky, isotropic stochastic, and directional searches discussed here. Each limit must state whether it is a Bayesian credible bound, a frequentist confidence bound, or an injection-defined sensitivity. None is a measured positive signal.

Helpful background. Continuous-Wave Phase Models supplies the source-side timing model. Power Spectral Density and Nonstationarity and Antenna Patterns and Polarization Response supply the noise and projection conventions.

Enter this chapter

The first two pages follow deterministic phase; the next four progressively relax the stochastic assumptions.

1. Targeted Continuous-Wave Searches maps an electromagnetic ephemeris into barycentric gravitational-wave phase, derives coherent mismatch, and specifies clock, orbit, glitch, and injection checks.
2. All-Sky Continuous-Wave Searches replaces the known ephemeris with a metric template bank and a finite-cost semicoherent hierarchy. It connects sensitivity, template density, trials, lines, and longer-coherence follow-up.
3. Cross-Correlation for Stochastic Backgrounds derives the overlap reduction function and optimal filter that reveal common strain below individual detector noise.
4. Anisotropic Stochastic Mapping expands the sky in pixels or harmonics, constructs the dirty map and Fisher matrix, and exposes regularization and null-space dependence.
5. Polarized and Non-Gaussian Backgrounds adds circular polarization, alternative responses, and intermittent populations, emphasizing network rank and noise-tail alternatives.
6. Unresolved Foregrounds and Confusion treats resolvability as analysis dependent and studies occupancy, subtraction residuals, and astrophysical–cosmological component degeneracy.

Each page provides a solved calculation and a computational or statistical failure test. Together they connect the ideal estimator to the released observational statement.

Two kinds of accumulated evidence

For a deterministic signal with accurately tracked phase, the coherent matched-filter signal-to-noise ratio scales schematically as

$$\rho_{\rm coh} \propto h_0\sqrt{\frac{T}{S_n}}.$$

An all-sky search cannot usually retain this coherence over the full run at acceptable cost; semicoherent methods trade sensitivity for parameter-space coverage. Phase error, not merely amplitude noise, is the decisive failure mode.

For a stationary stochastic field, one baseline instead measures

$$\left\langle \widetilde s_1^*(f)\widetilde s_2(f') \right\rangle = \frac12\delta(f-f') \gamma_{12}(f)S_h(f),$$

provided correlated detector noise is absent or controlled. The overlap function $\gamma_{12}$ carries geometry. Anisotropy, polarization, and multiple spectral components turn this scalar relation into a linear inverse problem whose small singular values identify poorly measured combinations.

The difference in scaling is useful: coherent searches estimate a strain amplitude, whereas cross-correlation estimates strain power. In the weak-background limit, the stochastic statistic has $\rho\propto S_h\sqrt T$. With fixed spectral shape $S_h\propto h_c^2$, a fixed-threshold characteristic-strain sensitivity therefore scales as $T^{-1/4}$, even though the uncertainty on the power amplitude falls as $T^{-1/2}$.

The methods meet when a population is neither individually resolved nor fully Gaussian. Segment-level mixture likelihoods and joint source-plus-background analyses interpolate between coherent events and a two-point stochastic description.

Inputs and checks for computation

Every long-duration analysis should pin:

• detector segments, calibration, clocks, ephemerides, data-quality decisions, and spectral notches;
• Fourier normalization, window, overlap, frequency resolution, and treatment of data gaps;
• signal parameterization, priors, template or sky basis, trials, and veto sequence;
• software version, random seeds, floating-point choices, convergence criteria, and computing cost;
• injection recovery, zero-signal coverage, time or sky shifts, line tests, and environmental witnesses.

Convergence must address the estimator, not only program execution. A coherent tone should recover the predicted sinc mismatch under a known frequency offset. A correlated Gaussian injection should recover its amplitude and variance as segment length and binning are refined. A sky injection should reproduce the point-spread function and reveal the Fisher null space. An intermittent foreground should move smoothly between mixture and Gaussian regimes as duty cycle changes.

Reading the cited null results

Search class	Current release at the cutoff	Correct interpretation
Known pulsars	LVK O4a targeted search	No credible detection; source- and orientation-dependent amplitude limits.
Isolated all-sky sources	LVK O4a all-sky search	No credible detection in the searched phase space; injection-calibrated upper limits.
Isotropic stochastic background	LVK O4a isotropic search	No significant background; limits conditional on spectral model and analyzed network.
Directional persistent background	LVK O4a directional search	No significant sky signal; map and radiometer limits conditional on morphology, basis, and regularization.

These releases do not exclude signals outside their phase evolution, frequency band, polarization, duty cycle, or population model. Refresh the table when official observing data, calibration, or analysis products supersede the cited versions.

Review the chapter

1. Why does a longer coherent time simultaneously help sensitivity and hurt all-sky computing cost?

Solution

Coherent signal-to-noise grows as $\sqrt{T_{\rm coh}}$, but parameter offsets accumulate more phase. Frequency, spindown, and sky-Doppler cells therefore shrink, increasing the template count. All-sky optimization chooses $T_{\rm coh}$ by balancing these trends under a fixed cost and false-dismissal target.

2. What distinguishes an overlap-function zero from a frequency at which no background exists?

Solution

At an overlap zero, sky contributions cancel for that detector pair, so the pair has no response to the assumed background even if $S_h(f)$ is nonzero. Another baseline with different geometry may retain sensitivity. It is a property of the instrument response, not the source spectrum.

3. Why can subtracting every individually detected source bias a residual-background search?

Solution

Estimated source parameters have correlated uncertainty, and waveforms and calibration are imperfect. Point subtraction leaves coherent residuals shaped by those errors. Near threshold, selection further biases which sources are removed. A joint or posterior-marginalized analysis propagates this uncertainty instead of treating residuals as ordinary stationary noise.

References

• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025). “Directional Search for Persistent Gravitational Waves: Results from the First Part of LIGO–Virgo–KAGRA’s Fourth Observing Run.” arXiv:2510.17487.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025). “Search for continuous gravitational waves from known pulsars in the first part of the fourth LIGO–Virgo–KAGRA observing run.” arXiv:2501.01495.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2025). “Upper Limits on the Isotropic Gravitational-Wave Background from the first part of LIGO, Virgo, and KAGRA’s fourth Observing Run.” arXiv:2508.20721.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “All-sky Searches for Continuous Gravitational Waves from Isolated Neutron Stars in the Data from the First Part of the Fourth LIGO–Virgo–KAGRA Observing Run.” arXiv:2603.14168.

Further reading

• Riles, K. (2023). “Searches for continuous-wave gravitational radiation.” Living Reviews in Relativity 26, 3. DOI.
• Romano, J. D., and Cornish, N. J. (2017). “Detection methods for stochastic gravitational-wave backgrounds: a unified treatment.” Living Reviews in Relativity 20, 2. DOI.